On base sizes for almost simple primitive groups

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Let G⩽Sym(Ω) be a finite almost simple primitive permutation group with socle G0. A subset of Ω is a base for G if its pointwise stabilizer is trivial; the base size of G, denoted b(G), is the minimal size of a base. We say that G is standard if G0=An and Ω is an orbit of subsets or partitions of {1,…,n}, or if G0 is a classical group and Ω is an orbit of subspaces (or pairs of subspaces) of the natural module for G0. The base size of a standard group can be arbitrarily large, in general, whereas the situation for non-standard groups is rather more restricted. Indeed, we have b(G)⩽7 for every non-standard group G, with equality if and only if G is the Mathieu group M24 in its natural action on 24 points. In this paper, we extend this result by classifying the non-standard groups with b(G)=6. The main tools include recent work on bases for actions of simple algebraic groups, together with probabilistic methods and improved fixed point ratio estimates for exceptional groups of Lie type.

Original languageEnglish
Pages (from-to)38-74
Number of pages37
JournalJournal of Algebra
Early online date11 Sep 2018
Publication statusPublished - 15 Dec 2018


  • Primitive permutation groups
  • base sizes
  • simple groups


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